Finite Group Databasebeta

← C7×#7760 #13184976

Subgroups — C7×#7760 #13184976

Order 1792.

424 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2809C211yesmaximal, characteristic
2C7×#2058C212yesmaximal
3C7×#2016C211yesmaximal, characteristic
4C7×#1272C211yesmaximal, characteristic
5C7×#1271C212yesmaximal
6C22×C7×Q16—21no—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C7×#106—22no—
9C2×C7×#103—21no—
10C7×#400—21no—
11C7×#398—22no—
12C7×#391—21no—
13C7×#391C2211yescharacteristic
14C7×#389C2212yes—
15C7×#387—22no—
16C7×#340—22no—
17C7×#336C2211yescharacteristic
18C7×#336C2212yes—
19#7760C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D411yescharacteristic
22C2×C7×Q16—42no—
23C2×C7×Q16—42no—
24C2×C7×Q16—42no—
25C22×C7×C8—22no—
26C22×C7×C8—21no—
27C7×C4⋊Q8—42no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D412yes—
33C2×C7×C4⋊C4D412yes—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C42×C7—21no—
36C2×C42×C7C2311yescharacteristic
37C7×#106—44no—
38C7×#103—42no—
39C7×#95—21no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44#2809C2×C711yescharacteristic
45#2058C2×C712yes—
46#2016C2×C711yescharacteristic
47#1272C2×C711yescharacteristic
48#1271C2×C712yes—
49C2×C7×Q8—21no—
50C2×C7×Q8—22no—
51C2×C7×Q8—41no—
52C2×C7×Q8—41no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C2×C7×Q8SD1612yes—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D412yes—
66C7×Q16—48no—
67C2×C7×C8—22no—
68C2×C7×C8—24no—
69C2×C7×C8—42no—
70C2×C7×C8—41no—
71C2×C7×C8—41no—
72C2×C7×C8—42no—
73C7×C4⋊C4—24no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—81no—
82C7×C4⋊C4—81no—
83C42×C7—41no—
84C42×C7—81no—
85C22×Q16—21no—
86C2×C4⋊Q8C22×C711yescharacteristic
87C2×#106—22no—
88C2×#103—21no—
89#400—21no—
90#398—22no—
91#391—21no—
92#391C22×C711yescharacteristic
93#389C22×C712yes—
94#387—22no—
95#340—22no—
96#336C22×C712yes—
97#336C22×C711yescharacteristic
98C23×C7#12011yescharacteristic
99C7×Q8—42no—
100C7×Q8—42no—
101C7×Q8—42no—
102C7×Q8—81no—
103C7×Q8—81no—
104C7×Q8—81no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—41no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—81no—
122C2×C4×C7—81no—
123C2×C4×C7C2×SD1611yescharacteristic
124C2×C4×C7#13512yes—
125C2×C4×C7#13711yescharacteristic
126C2×C4×C7#13712yes—
127C7×C8—44no—
128C7×C8—42no—
129C22×Q8—21no—
130C22×Q8C7×D411yescharacteristic
131C2×Q16—42no—
132C2×Q16—42no—
133C2×Q16—42no—
134C22×C8—21no—
135C22×C8—22no—
136C4⋊Q8—42no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C7×D412yes—
142C2×C4⋊C4C7×D412yes—
143C2×C4⋊C4C7×D411yescharacteristic
144C2×C42—21no—
145C2×C42C23×C711yescharacteristic
146#106—44no—
147#103—42no—
148#95—21no—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—22no—
153C22×C7#44811yescharacteristic
154C22×C7#45212yes—
155C22×C7#45311yescharacteristic
156C22×C7#45612yes—
157C22×C7B(4,𝔽2)11yescharacteristic
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—42no—
163C4×C7—82no—
164C4×C7—82no—
165C4×C7—82no—
166C4×C7—81no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C7×SD1612yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4C2×C7×D411yescharacteristic
183C22×C4C2×C7×D412yes—
184Q16—48no—
185C2×C8—22no—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C42—41no—
202C42—81no—
203C2×C7#159611yescharacteristic
204C2×C7#160312yes—
205C2×C7#163411yescharacteristic
206C2×C7#165312yes—
207C2×C7#166011yescharacteristic
208C23C7×#12011yescharacteristic
209Q8—42no—
210Q8—42no—
211Q8—42no—
212Q8—81no—
213Q8—81no—
214Q8—81no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4C2×C7×SD1611yescharacteristic
234C2×C4C7×#13512yes—
235C2×C4C7×#13711yescharacteristic
236C2×C4C7×#13712yes—
237C8—44no—
238C8—42no—
239C7#776011yescharacteristic, direct product factor, Sylow
240C22C7×#44811yescharacteristic
241C22C7×#45212yes—
242C22C7×#45311yescharacteristic
243C22C7×#45612yes—
244C22C7×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—82no—
251C4—82no—
252C4—82no—
253C4—81no—
254C2C7×#159611yescharacteristic
255C2C7×#160312yes—
256C2C7×#163411yescharacteristic
257C2C7×#165312yes—
258C2C7×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
19. #7760C7direct (N × H)
239. C7#7760direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#28092
C7×#20582
C7×#20162
C7×#12722
C7×#12712
#77607